156 ANALYTICAL MECHANICS axis of a point where if all the mass of the body were concentrated its moment of inertia would not change. Let m denote the mass of a body, 7 its moment of inertia with respect to a given axis, and K its radius of gyration relative to the same axis; then the definition gives I=K2m, or J5> V- (V) If Kc denote the radius of gyration relative to a parallel axis through the center of mass, then by equations (IV) and (V) we obtain K***K*+a*. (VI) ILLUSTRATIVE EXAMPLES. 1. Find the moment of inertia of a homogeneous circular disk (a) about its geometrical axis, (b) about one of the elements of its lateral surface. Let m be the mass, a the radius, I the thickness, and r the density of the disk. Then choosing a circular ring for the element of mass we have dm = T • I • 2 irr • dr, where r is the radius of the ring and dr its thickness. Therefore the moment of inertia about the axis of the disk is rlira4 FIG. 87. The moment of inertia about the element is obtained easily by the help of theorem II. Thus I' = / + ma2 It will be noticed that the thickness of the disk does not enter into the expressions for I and /' except through the mass of the disk. Therefore these expressions hold good whether the disk is thick enough to be called a cylinder or thin enough to be called a circular lamina.